Some more generalized integration operators on general function spaces

Zhi-jie Jiang; Zhi-jie Jiang

doi:10.3934/math.2026663

AIMS Mathematics

2026, Volume 11, Issue 6: 16129-16143. doi: 10.3934/math.2026663

Previous Article Next Article

Research article

Some more generalized integration operators on general function spaces

Zhi-jie Jiang ^,

School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, China

Received: 15 February 2026 Revised: 19 May 2026 Accepted: 25 May 2026 Published: 05 June 2026
MSC : 30H20, 47B38

Let $ H({\mathbb D}) $ be the class of all analytic functions on $ \mathbb{D} $ and $ \vec{h} = (h_0, h_1, \ldots, h_{n-1}) $ with $ h_k\in H({\mathbb D}) $ for $ k = 0, 1, \ldots, n-1 $. In this paper, we study a complex integration operator
$ \begin{align*} (T^{(n)}_{\vec{h}}f)(z) = I^n\big(h^{(n)}_0f+h^{(n-1)}_{1}f'+\cdots+h_{n-1}'f^{(n-1)}\big)(z), \, \, \, f\in H( {\mathbb D}), \end{align*} $
where $ I $ is the classical integration operator
$ (If)(z) = \int_{0}^zf(w)dw $
and $ I^n $ is the $ n $th iteration of $ I $. We characterize the bounded and compact operators $ T_{\vec{h}}^{(n)} $ on $ F(p, q, s) $ by using some characterizations of the space.
- $ F(p, q, s) $ space,
- integration operator,
- boundedness,
- compactness
Citation: Zhi-jie Jiang. Some more generalized integration operators on general function spaces[J]. AIMS Mathematics, 2026, 11(6): 16129-16143. doi: 10.3934/math.2026663

Related Papers:

Abstract

Let $ H({\mathbb D}) $ be the class of all analytic functions on $ \mathbb{D} $ and $ \vec{h} = (h_0, h_1, \ldots, h_{n-1}) $ with $ h_k\in H({\mathbb D}) $ for $ k = 0, 1, \ldots, n-1 $. In this paper, we study a complex integration operator

$ \begin{align*} (T^{(n)}_{\vec{h}}f)(z) = I^n\big(h^{(n)}_0f+h^{(n-1)}_{1}f'+\cdots+h_{n-1}'f^{(n-1)}\big)(z), \, \, \, f\in H( {\mathbb D}), \end{align*} $

where $ I $ is the classical integration operator

$ (If)(z) = \int_{0}^zf(w)dw $

and $ I^n $ is the $ n $th iteration of $ I $. We characterize the bounded and compact operators $ T_{\vec{h}}^{(n)} $ on $ F(p, q, s) $ by using some characterizations of the space.

References

[1]	A. Aleman, A. Siskakis, An integral operator on $H^p$, J. Funct. Anal., 28 (1995), 149–158. http://dx.doi.org/10.1080/17476939508814844 doi: 10.1080/17476939508814844
[2]	A. Aleman, A. Siskakis, Integration operators on Bergman spaces, Indiana U. Math. J., 46 (1997), 337–356. http://dx.doi.org/stable/24899619
[3]	H. Arroussi, H. He, C. Tong, X. Yang, Z. Yang, A new class of Carleson measures and integral operators on Fock spaces, Mediterr. J. Math., 22 (2025), 22. http://dx.doi.org/10.1007/s00009-024-02785-z doi: 10.1007/s00009-024-02785-z
[4]	N. Chalmoukis, Generalized integration operators on Hardy spaces, P. Am. Math. Soc., 148 (2020), 3325–3337. http://dx.doi.org/1909.00636
[5]	N. Chalmoukis, G. Nikolaidis, On the boundedness of generalized integration operators on Hardy spaces, Collect. Math., 77 (2026), 195–213. http://dx.doi.org/10.1007/s13348-024-00464-6 doi: 10.1007/s13348-024-00464-6
[6]	J. A. Cima, The basic properties of Bloch functions, Int. J. Math. Sci., 2 (1979), 369–413. http://dx.doi.org/10.1155/S0161171279000314 doi: 10.1155/S0161171279000314
[7]	P. Galanopoulos, D. Girela, J. A. Peláez, Multipliers and integration operators on Dirichlet spaces, T. Am. Math. Soc., 363 (2011), 1855–1886. http://dx.doi.org/10.1090/S0002-9947-2010-05137-2 doi: 10.1090/S0002-9947-2010-05137-2
[8]	D. Girela, J. A. Peláez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal., 214 (2006), 334–358. http://dx.doi.org/10.1016/j.jfa.2006.04.025 doi: 10.1016/j.jfa.2006.04.025
[9]	W. Hastings, A Carleson measure theorem for Bergman spaces, P. Am. Math. Soc., 214 (2006), 334–358. http://dx.doi.org/10.1016/j.jfa.2006.04.025 doi: 10.1016/j.jfa.2006.04.025
[10]	Z. J. Jiang, X. F. Wang, Products of radial derivative and weighted composition operators from weighted Bergman-Orlicz spaces to weighted-type spaces, Oper. Matrices, 12 (2018), 301–319. http://dx.doi.org/10.7153/oam-2018-12-20 doi: 10.7153/oam-2018-12-20
[11]	Z. J. Jiang, Product-type operators from Zygmund spaces to Bloch-Orlicz spaces, Complex. Var. Elliptic., 62 (2017), 1645–1664. http://dx.doi.org/10.1080/17476933.2016.1278436 doi: 10.1080/17476933.2016.1278436
[12]	S. Li, S. Stević, Product-type operators from logarithmic Bergman-type spaces to Zygmund-Orlicz spaces, Appl. Math. Comput., 215 (2009), 464–473. http://dx.doi.org/10.1016/j.amc.2009.05.011 doi: 10.1016/j.amc.2009.05.011
[13]	S. Li, S. Stević, Integral type operators from mixed-norm spaces to ${\alpha}$-Bloch spaces, Integr. Transf. Spec. F., 18 (2007), 485–493. http://dx.doi.org/10.1080/10652460701320703 doi: 10.1080/10652460701320703
[14]	S. Ohno, K. Stroethoff, R. Zhao, Weighted composition operators between Bloch-type spaces, Rocky. Mt. J. Math., 33 (2003), 191–215. http://dx.doi.org/stable/44238919
[15]	R. Qian, X. Zhu, Embedding Hardy spaces $H^p$ into tent spaces and generalized integration operators, Ann. Pol. Math., 128 (2022), 143–157. http://dx.doi.org/10.4064/ap210512-1-10 doi: 10.4064/ap210512-1-10
[16]	J. Rättyä, $n$-th derivative characterizations, mean growth of derivatives and $F(p, q, s)$, B. Aust. Math. Soc., 68 (2003), 405–421. http://dx.doi.org/10.4064/ap210512-1-10 doi: 10.4064/ap210512-1-10
[17]	H. J. Schwartz, Composition operators on $H^p$, Toledo, Ohio: The University of Toledo, 1969.
[18]	Y. Shi, S. Li, Linear combination of composition operators on $H^\infty$ and the Bloch space, Arch. Math., 112 (2019), 511–519. http://dx.doi.org/1802.04092
[19]	S. Stević, Composition followed by differentiation from $H^\infty$ and the Bloch space to $n$th weighted-type spaces on the unit disk, Appl. Math. Comput., 216 (2010), 3450–3458. http://dx.doi.org/10.1016/j.amc.2010.03.117 doi: 10.1016/j.amc.2010.03.117
[20]	S. Stević, Composition operators from the Hardy space to the $n$th weighted-type space on the unit disk and the half-plane, Appl. Math. Comput., 215 (2010), 3950–3955. http://dx.doi.org/10.1016/j.amc.2009.11.043 doi: 10.1016/j.amc.2009.11.043
[21]	S. Stević, Composition operators from the weighted Bergman space to the $n$th weighted spaces on the unit disc, Discrete. Dyn. Nat. Soc., 2009 (2009), 742019. http://dx.doi.org/10.1155/2009/742019 doi: 10.1155/2009/742019
[22]	S. Stević, Boundedness and compactness of an integral operator on mixed norm spaces on the polydisc, Siberian. Math. J., 48 (2007), 559–569. http://dx.doi.org/10.1007/s11202-007-0058-5 doi: 10.1007/s11202-007-0058-5
[23]	S. Stević, Boundedness and compactness of an integral operator on a weighted space on the polydisc, Indian. J. Pure Ap. Mat., 37 (2006), 343–355.
[24]	R. Zhao, On $F(p, q, s)$ spaces, Acta. Math. Sci., 41 (2021), 1985–2020. http://dx.doi.org/10.1007/s10473-021-0613-3 doi: 10.1007/s10473-021-0613-3
[25]	R. Zhao, On a general family of function spaces, Helsinki: Suomalainen tiedeakatemia, 1996.
[26]	X. Zhu, D. Qiu, Generalized integration operators from the Besov space into general function spaces, J. Math. Inequal., 19 (2025), 151–163. http://dx.doi.org/10.7153/jmi-2025-19-10 doi: 10.7153/jmi-2025-19-10
[27]	K. Zhu, Bloch type spaces of analytic functions, Rocky. Mt. J. Math., 23 (1993), 1143–1177. http://dx.doi.org/stable/44237763
[28]	K. Zhu, Operator theory in function spaces, 2 Eds., New York: Marecl Dekker, 1990. http://dx.doi.org/10.1016/0378-4754(91)90069-f

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)